You and I play a game involving successive throws of a fair coin.
Suppose I pick HH and you pick TH. The coin is thrown repeatedly
until we see either two heads in a row (I win) or a tail followed
by a head (you win). What is the probability that you win?

Thank Your Lucky Stars

Stage: 4 Challenge Level:

Thank you for this solution Andrei (Andrei
Lazanu, School 205 Bucharest) and for the link to the useful
site:

For a 2x2 grid I need to
make 2 moves.

There are 4 possible paths:

left, left
left, up
up, left
up, up

Only 2 of these take me to the top left-hand corner of the grid, so
the probability of getting to the opposite corner is: $$ {{1
\over{2^2}}\times2} = {2 \over4} = {1 \over2} $$

For a 3x3 grid I need to
make 4 moves.

There are 16 possible paths.

Only 6 of these take me to the top left-hand corner of the grid, so
the probability of getting to the opposite corner is: $$ {{1
\over{2^4}}\times6} = {6 \over16} = {3 \over8} $$

For a 4x4 grid I need to
make 6 moves.

There are 64 possible paths.

Only 20 of these take me to the top left-hand corner of the
grid, so the probability of getting to the opposite corner is: $$
{{1 \over{2^6}}\times20} = {20 \over64} = {5 \over16} $$

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